I get it. DD or other dd utils don't work to boot. Not TC's fault, but just the way it's done, and modern uefi machinery can sometimes present issues.
UNOFFICIAL way to kick the tires without going bananas about hardware etc:
Create a VENTOY usb stick. Use the latest version as they have fixed a TC boot issue in 1.104. Both windows and linux cli versions available. This will boot the iso directly.
Put your tinycore iso on the Ventoy stick and reboot.
Once booted into TC, run from the terminal:
tce-setdrive
and point this to any writable disk you have.
Now you can play around, have persistence, add applications, drivers, utils, and so forth. Treat this Ventoy boot stick like a CD from the olden days (because it is booting a read-only iso) - that is, if you want to make any changes to your TC grub boot config for specific options, you'll need to temporarily halt the process, typically with the "E" key. Make your necessary changes, and continue to boot with F10. It's there, you'll see.
At least this will get you into a working environment with persistence and get familiar from a bootable usb stick viewpoint before delving into much more complicated setups.